130 (2005)
MATHEMATICA BOHEMICA
No. 1, 1–18
A NEVANLINNA THEOREM FOR SUPERHARMONIC FUNCTIONS
ON DIRICHLET REGULAR GREENIAN SETS
, Christchurch
(Received January 13, 2004)
Abstract. A generalization of Nevanlinna’s First Fundamental Theorem to superharmonic
functions on Green balls is proved. This enables us to generalize many other theorems,
on the behaviour of mean values of superharmonic functions over Green spheres, on the
Hausdorff measures of certain sets, on the Riesz measures of superharmonic functions, and
on differences of positive superharmonic functions.
Keywords: Nevanlinna theorem, superharmonic function, δ-subharmonic function, Riesz
measure, mean value
MSC 2000 : 31B05, 31B10
1. Introduction
Nevanlinna’s First Fundamental Theorem is concerned with superharmonic functions on balls, and has applications to superharmonic functions on n and δsubharmonic functions on balls [5], [6]. Here we prove a generalization to superharmonic functions on Green balls, which are sets of the form
BD (x0 , r) = {y ∈ D : GD (x0 , y) > τ (r)},
where D is a Dirichlet regular Greenian open set, GD is its Green function, τ (r) =
− log r if n = 2, τ (r) = r 2−n if n > 3, and 0 < r < 1 if n = 2, 0 < r < ∞ if n > 3. Any
Green ball BD (x0 , r) is a bounded domain with its closure in D [11], and is Dirichlet
regular. Our result involves the mean values of superharmonic functions over Green
spheres, introduced in [11]. Easy corollaries generalize results of Armitage [2], Kuran
[7], and Parker [8].
The theorem leads to generalizations of several other results, including some on
the behaviour of quotients of differences of mean values of δ-subharmonic functions
1
in [14], on the size of the sets where certain singularities occur in [13], on conditions
for a positive measure to be the Riesz measure for a superharmonic function with a
harmonic minorant in [6] p. 128, and on conditions for a δ-subharmonic function to
be expressible as a difference of two positive superharmonic functions in [5] p. 510.
Our results are analogues of theorems on supertemperatures given in [15].
We note that Armitage [1] has given a Nevanlinna theorem for superharmonic
functions on half-spaces, but his approach is not related to ours.
For all x, y ∈ n , we put G(x, y) = τ (kx − yk) and
B(x, r) = {y : G(x, y) > τ (r)} = {y : kx − yk < r}
for all r > 0. We also put pn = max{1, n − 2}, and note that τ 0 (r) = −pn r1−n for
all n > 2.
For almost every r such that τ (r) > 0, the set {y ∈ D : GD (x0 , y) = τ (r)} is a
smooth regular (n − 1)-dimensional manifold. Such a value of r is called regular. If r
is a regular value, then the set is the Green sphere ∂BD (x0 , r), and the surface mean
value LD of a function u is defined by
Z
1
LD (u, x0 , r) =
k∇GD (x0 , ·)ku dσ
pn σn ∂BD (x0 ,r)
whenever the integral exists. Here σn denotes the surface area of the unit ball in n ,
and σ denotes surface area measure. If GD is replaced by G, then the formula for
LD (u, x0 , r) reduces to the standard formula for the mean value of u over the sphere
∂B(x0 , r), which we denote by L(u, x0 , r).
2. The generalization of Nevanlinna’s first fundamental theorem
In this section we present our generalization of formula (3.9.6) in [6]. We also
present some immediate consequences, which generalize, and to some extent unify,
results of Armitage [2], Kuran [7] and Parker [8].
Theorem 1. Let E be an open set, let D be a Dirichlet regular Greenian open
superset of E, let x0 ∈ E, and let r and s be regular values such that 0 < r < s and
B D (x0 , s) ⊆ E. If u is superharmonic on E with Riesz measure µ, then
Z s
(1)
LD (u, x0 , r) = LD (u, x0 , s) + pn
t1−n µ(B D (x0 , t)) dt
r
and
(2)
2
u(x0 ) = LD (u, x0 , s) + pn
Z
s
t1−n µ(B D (x0 , t)) dt.
0
.
Let V be a bounded open set such that B D (x0 , s) ⊆ V and V ⊆ E.
Then there is a harmonic function h such that
u = G D µV + h
on V . Recall that, by [11] Theorem 1, the means LD are finite-valued and
LD (h, x0 , r) = h(x0 ). It follows that
LD (u, x0 , r) − LD (u, x0 , s) = LD (GD µV , x0 , r) − LD (GD µV , x0 , s)
Z
(LD (GD (·, y), x0 , r) − LD (GD (·, y), x0 , s)) dµ(y)
=
ZV
=
((GD (x0 , y) ∧ τ (r)) − (GD (x0 , y) ∧ τ (s))) dµ(y)
V
by [12] Theorem 2. By definition of BD (x0 , r), we have GD (x0 , y) ∧ τ (r) = τ (r) if
and only if y ∈ B D (x0 , r). Therefore
(GD (x0 , y) ∧ τ (r)) − (GD (x0 , y) ∧ τ (s))

τ (r) − τ (s)


= GD (x0 , y) − τ (s)


0
if y ∈ B D (x0 , r),
if y ∈ B D (x0 , s) \ B D (x0 , r),
if y ∈
/ B D (x0 , s).
Hence
LD (u, x0 , r) − LD (u, x0 , s) =
Z
B D (x0 ,s)
((GD (x0 , y) ∧ τ (r)) − τ (s)) dµ(y).
If we now put λ(t) = µ(B D (x0 , t)) whenever 0 6 t 6 s, we obtain
Z
LD (u, x0 , r) − LD (u, x0 , s) =
((τ (t) ∧ τ (r)) − τ (s)) dλ(t)
[0,s]
= (τ (r) − τ (s))λ(0) + ((τ (t) ∧ τ (r)) − τ (s))λ(t)|s0+
Z s
−
τ 0 (t)λ(t) dt
r
Z s
t1−n µ(B D (x0 , t)) dt.
= pn
r
This proves (1). Making r → 0 in (1), we obtain (2).
!"$#
. The formula (2) is a direct extension of Nevanlinna’s first fundamental theorem ([6] p. 127). If n = 2, consider the case E = D = B(0, r0 ) and x0 = 0.
Then, whenever 0 < r < 1, we have
BD (x0 , r) = {x ∈ B(0, r0 ) : GB(0,r0 ) (0, x) > τ (r)} = B(0, rr0 )
3
and
LD (u, x0 , r) = κ−1
2
Z
∂BD (0,r)
= κ−1
2
Z
∂B(0,rr0 )
k∇GB(0,r0 ) (0, ·)ku dσ
1
u dσ = L(u, 0, rr0 ),
kxk
so that (2) becomes
u(0) = L(u, 0, sr0 )+pn
Z
s
t
−1
µ(B(0, tr0 )) dt = L(u, 0, sr0 )+pn
0
Z
sr0
t−1 µ(B(0, t)) dt
0
for 0 < s < 1. On the other hand, if n > 3 we can take E = B(0, r0 ), x0 = 0, and
D = n . Then, whenever 0 < r < ∞, we have BD (x0 , r) = B(0, r) and
Z
LD (u, x0 , r) = κ−1
(2 − n)kxk1−n u dσ = L(u, 0, r),
n
∂B(0,r)
so that we can obtain the classical formula from (2) by removing the subscripts D.
Similarly, formula (1) extends a variant of the classical result given, for example,
in [2] Lemma 3.
If we put
Z s
ND (x0 , s) = pn
t1−n µ(B D (x0 , t)) dt,
0
then ND (x0 , ·) is obviously increasing, and a standard argument ([6] p. 127) shows
that there is a convex function ϕ such that ND (x0 , ·) = ϕ ◦ τ.
We now give three corollaries of Theorem 1, all of which are extensions of known
results. Theorem 3 (iv), (v) of [11] imply that the surface means can be replaced by
the corresponding volume means in the first two corollaries.
Corollary 1. Let E be an open set, let D be a Dirichlet regular Greenian open
superset of E, and let x0 ∈ E. If u is superharmonic on E with Riesz measure µ,
then as r → 0 through regular values
(3)
lim
LD (u, x0 , r)
= µ({x0 }).
τ (r)
.
Given ε > 0, choose δ > 0 such that |µ(B D (x0 , t)) − µ({x0 })| < ε for
all t < δ. Fix a regular value of s < δ. Then, whenever r is a regular value and r < s,
we have
Z s
(4)
LD (u, x0 , r) = LD (u, x0 , s) + pn
t1−n µ(B D (x0 , t)) dt
r
4
by (1). Since s < δ, as r → 0 we have
pn
τ (r)
Z
s
t1−n µ(B D (x0 , t)) dt <
r
so that
lim sup
pn
τ (r)
Z
pn
τ (r)
Z
(µ({x0 }) + ε)(τ (r) − τ (s))
→ µ({x0 }) + ε,
τ (r)
s
r
t1−n µ(B D (x0 , t)) dt 6 µ({x0 }) + ε.
Similarly
lim inf
s
r
t1−n µ(B D (x0 , t)) dt > µ({x0 }) − ε,
so that the corresponding limit exists and is µ({x0 }). The result (3) now follows from
(4).
The cases D = n with n > 3, and D a ball centred at x0 with n = 2, of Corollary 1
were proved by Parker ([8] Lemma). Earlier, Armitage ([2] Lemma 3 Corollary 1)
had proved (for the same cases) that if either side of (3) is zero then so is the other,
and Kuran ([7] Theorem 2) had proved that if u(x0 ) = 0 then µ({x0 }) = 0.
Corollary 2. Let D be a Dirichlet regular Greenian open set, let x0 ∈ D, and let
u be superharmonic with Riesz measure µ on D. If u > 0, then
τ (r)µ(B D (x0 , r)) 6 LD (u, x0 , r)
(5)
for all regular values of r.
.
Let r and s be regular values such that r < s. Put R = 1 if n = 2, and
R = ∞ if n > 3. Then 0 < r < s < R, so that by (1)
LD (u, x0 , r) > pn
Z
s
t1−n µ(B D (x0 , t)) dt
r
> µ(B D (x0 , r))pn
Z
R
t1−n dt = µ(B D (x0 , r))τ (r).
r
!"$#
. The special case of Corollary 2 in which D = n and n > 3, was
proved earlier by Kuran ([7] Theorem 4) and Armitage ([2] Lemma 3 Corollary 2).
The proof given above follows that of Armitage.
5
The case where D = B(0, %0 ) and n = 2 of Corollary 2, implies the second inequality of [7] Theorem 4. For then, taking x0 = 0, we have BD (x0 , r) = B(0, r%0 ),
and LD (u, x0 , r) = L(u, 0, r%0 ), so that (5) becomes
log
1
µ(B(0, r%0 )) 6 L(u, 0, r%0 )
r
whenever 0 < r < 1. If we now put % = r%0 , and confine r to ]0, θ[ for some θ < 1,
we get
1 −1
1 −1
µ(B(0, %)) 6 log
L(u, 0, %) 6 log
L(u, 0, %),
r
θ
which is (13) of [7].
Corollary 3. Let D be a Dirichlet regular Greenian open set, and let u be
superharmonic with Riesz measure µ on D. Put R = 1 if n = 2, and R = ∞ if
n > 3. If u > 0, then
lim τ (r)µ(B D (x, r)) = 0
r→R
for all x ∈ D.
.
The greatest harmonic minorant h of u is given by
h(x) = lim LD (u, x, r)
r→R
for all x ∈ D, in view of [11] Theorem 1 and [3] p. 123, (11.1). Since µ is the Riesz
measure for u − h, it therefore follows from Corollary 2 above and [11] Theorem 1
that
τ (r)µ(B D (x, r)) 6 LD (u − h, x, r) = LD (u, x, r) − h(x) → 0
as r → R through regular values. Therefore, given ε > 0 we can find K such that
µ(B D (x, r)) 6 ε/τ (r) for all regular values of r > K, and hence for every r > K
because 1/τ is continuous and µ(B D (x, ·)) is an increasing function.
The case where D =
Theorem 1.
6
n
and n > 3 of Corollary 3 was first proved by Kuran [7]
3. The behaviour of the means for small regular values
Theorem 2 below generalizes part of [14] Theorem 2, which dealt only with the
classical spherical means.
We need some notation. Let E be an open set, and let D be a Dirichlet regular
Greenian open superset of E. If B D (x0 , s) ⊆ E, ν is a positive measure on E, and
0 6 r < s, we put
Z s
t1−n ν(B D (x0 , t)) dt.
Iν,D (x0 ; r, s) = pn
r
Theorem 2. Let E be an open set, let D be a Dirichlet regular Greenian open
superset of E, let u be δ-subharmonic on E with Riesz measure µ, and let ν be a
positive measure on E. Then
µ(B D (x, t))
LD (u, x, r) − LD (u, x, s)
6 lim sup
I
(x;
r,
s)
ν(B D (x, t))
t→0
0<r<s→0
ν,D
lim sup
whenever the latter exists. Furthermore, if u(x) is defined and finite, and Iν,D (x;
0, s) < ∞ for all sufficiently small values of s, then
lim sup
.
s→0
u(x) − LD (u, x, s)
µ(B D (x, t))
6 lim sup
.
Iν,D (x; 0, s)
ν(B D (x, t))
t→0
The proof of the first inequality is similar to the proof of the first part
of [14] Theorem 2. The proof of the second part is similar again, using (2) instead
of (1).
Theorem 2 can easily be rewritten in a form that generalizes [14] Theorem 6.
Theorem 3. Let E be an open set, and let D be a Dirichlet regular Greenian
open superset of E. Let u be δ-subharmonic with Riesz measure µ, and let v be
superharmonic with Riesz measure ν, on E. Then
lim sup
0<r<s→0
µ(B D (x, t))
LD (u, x, r) − LD (u, x, s)
6 lim sup
LD (v, x, r) − LD (v, x, s)
ν(B D (x, t))
t→0
whenever the latter exists. Furthermore, if u(x) is defined and finite, and v(x) < ∞,
then
u(x) − LD (u, x, s)
µ(B D (x, t))
lim sup
.
6 lim sup
v(x) − LD (v, x, s)
ν(B D (x, t))
s→0
t→0
.
In view of Theorem 1 and the finiteness of the means ([11] Theorem 1),
the result follows from Theorem 2.
We can also generalize [14] Theorem 5, as follows.
7
Theorem 4. Let E be an open set, let D be a Dirichlet regular Greenian open
superset of E, and let u be δ-subharmonic with Riesz measure µ on E. Let α > 0,
let f be a positive, increasing, absolutely continuous function on [0, α], and let
Z s
ˆ
t1−n f (t) dt
f (r, s) = pn
r
whenever 0 6 r < s 6 α. Then
LD (u, x, r) − LD (u, x, s)
µ(B D (x, t))
6 lim sup
ˆ
f (t)
0<r<s→0
t→0
f(r, s)
lim sup
for all x in E. Furthermore, if u(x) is defined and finite, and fˆ(0, s) < ∞ for all
sufficiently small values of s, then
lim sup
s→0
µ(B D (x, t))
u(x) − LD (u, x, s)
.
6 lim sup
ˆ
f (t)
t→0
f (0, s)
.
Given x, we choose % 6 α such that B D (x, %) ⊆ E, and define a positive
measure ν on E by putting
2
dν = −κ−1
n k∇GD (x, ·)k
f0 τ0
(τ −1 (GD (x, ·)))χBD (x,%) dλ + f (0) dδx ,
where τ −1 denotes the inverse function of τ, χA denotes the characteristic function
of a set A, λ denotes n-dimensional Lebesgue measure, and δx denotes the unit mass
at x. If 0 < t < %, it follows from results in [11] pp. 309–310 that
Z
f0 −1
k∇GD (x, ·)k2 0 (τ −1 (GD (x, ·))) dλ + f (0)
ν(B D (x, t)) = −κn
τ
BD (x,t)
Z tZ
0
= κ−1
k∇G
(x,
·)kf
(r)
dσ
dr + f (0)
D
n
0
=
Z
∂B D (x,r)
t
LD (1, x, r)f 0 (r) dr + f (0) =
0
Z
t
f 0 (r) dr + f (0) = f (t).
0
Therefore, whenever 0 6 r < s 6 %,
Iν,D (x; r, s) = pn
Z
s
t1−n f (t) dt = fˆ(r, s).
r
The results now follow from Theorem 2.
The corollaries of [14] Theorem 5 can now easily be generalized. We leave this to
the reader.
8
4. The Hausdorff measure of certain sets
We use Theorem 4 to study the size of the set of points x where
lim sup
0<r<s→0
LD (u, x, r) − LD (u, x, s)
fˆ(r, s)
is unbounded, or is positive, for a given function f and superharmonic function u.
The size is estimated in terms of Hausdorff measures [9]. Our results generalize
theorems of Armitage [2] and Watson [13] in two directions, namely the mean values
considered and the Hausdorff measures used.
Theorem 5. Let n > 3, let E be an open set, let D be a Dirichlet regular Greenian
open superset of E, and let u be superharmonic on E. Let h be an increasing,
absolutely continuous function on [0, ∞[ such that h(0) = 0 and h(2s) 6 Kh(s) for
all s > 0, where K is a constant. Put
ĥ(r, s) = pn
Z
s
t1−n h(t) dt.
r
Then the set
(6)
n
LD (u, x, r) − LD (u, x, s)
x ∈ E : lim sup
ĥ(r, s)
=∞
LD (u, x, r) − LD (u, x, s)
>0
0<r<s→0
o
has h-measure zero, and
(7)
n
x ∈ E : lim sup
0<r<s→0
ĥ(r, s)
o
is σ-finite with respect to h-measure.
.
Let µ denote the Riesz measure for u. It suffices to prove the results locally, and we may therefore suppose that E is bounded and µ is finite. By Theorem 4,
the set (6) is a subset of
(8)
n
x ∈ E : lim sup
t→0
o
µ(B D (x, t))
=∞ ,
h(t)
and the set (7) is contained in
(9)
n
x ∈ E : lim sup
t→0
o
µ(B D (x, t))
>0 .
h(t)
9
Since n > 3, GD (x, y) 6 kx − yk2−n for all x, y ∈ D, so that BD (x, t) ⊆ B(x, t) for
all t > 0. Therefore B D (x, t) is contained in a closed interval I(x, s) of centre x and
edge length s = 2r, which is contained in E if r is sufficiently small. It follows that
the set (8) is a subset of
n
(10)
o
µ(I(x, s))
=∞ ,
h(s/2)
x ∈ E : lim sup
s→0
and that the set (9) is a subset of
n
(11)
x ∈ E : lim sup
If i is chosen so that 2i−1 >
s→0
o
µ(I(x, s))
>0 .
h(s/2)
√
n, then
√
h(s/2) > K −i h(2i−1 s) > K −i h(s n) = K −i h(diam I(x, s)).
Therefore the sets (10) and (11) are contained in the sets
n
o
µ(I(x, s))
√
S = x ∈ E : lim sup
=∞
h(s n)
s→0
and
o
n
µ(I(x, s))
√
>0
T = x ∈ E : lim sup
h(s n)
s→0
respectively. By a result of Rogers and Taylor ([10], Lemma 2), for each k > 0 the
set
o
n
µ(I(x, s))
√
Sk = x ∈ E : lim sup
>k
h(s n)
s→0
has h-measure h − m(Sk ) 6 M µ(E)/k for some constant M. It follows that
h − m(S) = 0, and that T is σ-finite with respect to h-measure. This implies
the results of the theorem.
Corollary 1. Let n > 3, and let E, D, h and ĥ be as in Theorem 5. If u is
δ-subharmonic on E, and ĥ(0, α) = ∞ for some α, then the set
n
x ∈ E : lim sup
s→0
LD (u, x, s)
ĥ(s, α)
=∞
o
has h-measure zero, and
n
x ∈ E : lim sup
s→0
is σ-finite with respect to h-measure.
10
LD (u, x, s)
ĥ(s, α)
>0
o
.
Let µ denote the Riesz measure of u. It is enough to prove the result
locally, and so we may suppose that E is bounded (and hence Greenian) and that
µ has finite total variation. Since GE µ 6 GE |µ|, we may also suppose that µ is
positive (so that u is superharmonic). Using the method of proof of [14] Theorem 5
Corollary 1, we can now show that
lim sup
LD (u, x, s)
ĥ(s, α)
s→0
6 lim sup
0<r<s→0
LD (u, x, r) − LD (u, x, s),
ĥ(r, s)
and so the result follows from Theorem 5.
In the next result, we denote by mβ the h-measure constructed from the function
h(s) = sβ , where β > 0.
Corollary 2. Let n > 3, and let E, D and u be as in Corollary 1. Then the set
Sβ , defined by
n
o
Sβ = x ∈ E : lim sup sn−β−2 LD (u, x, s) = ∞
s→0
if 0 < β < n − 2, and by
n
o
1 −1
Sβ = x ∈ E : lim sup log
LD (u, x, s) = ∞
s
s→0
if β = n − 2, has mβ -measure zero. Furthermore, the set Tβ given by
n
o
Tβ = x ∈ E : lim sup sn−β−2 LD (u, x, s) > 0
s→0
if 0 < β < n − 2, and by
o
n
1 −1
LD (u, x, s) > 0
Tβ = x ∈ E : lim sup log
s
s→0
if β = n − 2, is σ-finite with respect to mβ .
.
If we take h(s) = sβ , 0 < β 6 n − 2, in Corollary 1, then ĥ(0, α) = ∞
so that the corollary is applicable. Furthermore,
pn
(sβ+2−n − αβ+2−n ) if 0 < β < n − 2,
n
−
β−2
ĥ(s, α) =

pn log(α/s)
if β = n − 2.


The result follows.
11
The case of Corollary 2 in which D = n , u is superharmonic, and 0 < β < n − 2,
was proved by Armitage in [2] Theorem 3. The case D = n was subsequently proved
by Watson in [13] Theorem 16 (in the statement of which |u| should be replaced by u).
In Theorem 5 Corollary 1, the condition on ĥ ensures that LD (u, x, s) → ∞ as
s → 0, for every x in either of the sets in question. Therefore the sets are polar. In
the theorem itself, polarity is not so readily determined, and in fact depends on h.
We demonstrate this in the context of the next corollary.
Corollary 3. Let n > 3, and let E, D and u be as in Theorem 5.
(i) If 0 < β < n − 2, then the set
n
o
LD (u, x, r) − LD (u, x, s)
=
∞
−(n−2−β) − s−(n−2−β)
0<r<s→0 r
x ∈ E : lim sup
has mβ -measure zero, and
n
o
LD (u, x, r) − LD (u, x, s)
>
0
−(n−2−β) − s−(n−2−β)
0<r<s→0 r
x ∈ E : lim sup
is σ-finite with respect to mβ .
(ii) The set
n
o
LD (u, x, r) − LD (u, x, s)
=∞
log(s/r)
0<r<s→0
x ∈ E : lim sup
has mn−2 -measure zero, and
n
o
LD (u, x, r) − LD (u, x, s)
>0
log(s/r)
0<r<s→0
x ∈ E : lim sup
is σ-finite with respect to mn−2 .
(iii) If n − 2 < β 6 n, then the set
n
o
LD (u, x, r) − LD (u, x, s)
=
∞
sβ+2−n − rβ+2−n
0<r<s→0
x ∈ E : lim sup
has mβ -measure zero, and
n
o
LD (u, x, r) − LD (u, x, s)
>0
β+2−n
β+2−n
s
−r
0<r<s→0
x ∈ E : lim sup
is σ-finite with respect to mβ .
.
12
Take h(s) = sβ , 0 < β 6 n, in Theorem 5.
Sets of finite mn−2 -measure are polar ([4] p. 78, or [6] p. 228). Therefore the sets in
Corollary 3 (i) and (ii) are all polar. The sets in (iii), however, need not be. Given β
such that n − 2 < β 6 n, choose γ such that n − 2 < γ < β. If S is an mγ -measurable
set for which 0 < mγ (S) < ∞, and µ is the restriction to S of mγ , then by [4] p. 25
we have
µ(B(x, t))
lim sup
>1
tγ
t→0
µ-a.e. on S. Therefore
lim sup
t→0
µ(B(x, t))
=∞
tβ
for all x ∈ S0 , say, where µ(S \ S0 ) = 0. It follows from Theorem 4 that
L(u, x, r) − L(u, x, s)
=∞
sβ+2−n − rβ+2−n
0<r<s→0
lim sup
for all x ∈ S0 . Since mγ (S0 ) > 0 and γ > n − 2, the set S0 is not polar ([4] p. 78, or
[6] p. 225), and so the sets in Corollary 3 (iii) are not polar.
5. The Riesz measures of superharmonic functions on
Dirichlet regular Greenian sets
In this section we generalize [6] Theorem 3.20 from the case where D = n for
some n > 3, to that where D is an arbitrary Dirichlet regular Greenian domain in
n for any n > 2.
Theorem 6. Let D be a Dirichlet regular Greenian domain, and let µ be a
positive measure on D. Put R = 1 if n = 2, and R = ∞ if n > 3.
(i) If µ is the Riesz measure of a superharmonic function that has a harmonic
minorant on D, then
(12)
Z
R
1
2
t1−n µ(B D (x, t)) dt < ∞
for all x ∈ D.
(ii) Conversely, if there is a point x ∈ D such that (12) holds, then GD µ is
superharmonic on D. If, in addition, µ({x}) = 0 and
Z
0
1
2
t1−n µ(B D (x, t)) dt < ∞,
then GD µ(x) < ∞.
13
.
(i) Let w be a superharmonic function which has a harmonic minorant
u on D, and whose Riesz measure is µ. Then µ is also the Riesz measure for w − u.
Therefore, if x ∈ D and r, s are regular values such that r < s, Theorem 1 shows
that
Z s
LD (w − u, x, r) = LD (w − u, x, s) + pn
t1−n µ(B D (x, t)) dt
r
Z s
t1−n µ(B D (x, t)) dt.
> pn
r
Since LD (w − u, x, r) < ∞ by [11] Theorem 1, if we fix r and make s → R we obtain
(12).
(ii) Now suppose that (12) holds for some x = x0 ∈ D. Let {kj } be an increasing
sequence of regular values such that kj → R as j → ∞, and put
AD (x0 ; k1 , R) = D \ B D (x0 , k1 ),
AD (x0 ; k1 , kj ) = BD (x0 , kj ) \ B D (x0 , k1 )
If u = GD µ, then for all x ∈ D we put
Z
Z
GD (x, y) dµ(y) +
u(x) =
B D (x0 ,k1 )
say, and
uj (x) =
Z
for all j > 1.
GD (x, y) dµ(y) = v1 (x) + v2 (x),
AD (x0 ;k1 ,R)
GD (x, y) dµ(y)
AD (x0 ;k1 ,kj )
for all j > 1. Since µ is locally finite, v1 and every uj is superharmonic on D. Since
{uj } is increasing to the limit v2 , if v2 (x0 ) < ∞ then v2 will be superharmonic on D.
Writing λ(t) = µ(B D (x0 , t)) for all t > 0, we have
uj (x0 ) =
Z
kj
k1
k
τ (t) dλ(t) = [τ (t)λ(t)]kj1 −
Z
kj
τ 0 (t)λ(t) dt.
k1
Since (12) holds when x = x0 , we have
λ(s)τ (s) = λ(s)
Z
R
s
−τ 0 (t) dt 6 −
Z
R
s
τ 0 (t)λ(t) dt → 0
as s → R. Therefore
v2 (x0 ) = lim uj (x0 ) = −τ (k1 )λ(k1 ) −
j→∞
so that v2 , and hence u, is superharmonic on D.
14
Z
R
k1
τ 0 (t)λ(t) dt < ∞,
For the last part, let {rj } be a decreasing null sequence of regular values (relative
to x0 ). Then, if µ({x0 }) = 0,
u(x0 ) = lim
j→∞
Z
R
τ (t) dλ(t) = lim
rj
j→∞
6−
Z
−τ (rj )λ(rj ) −
Z
R
τ 0 (t)λ(t) dt
rj
R
0
τ 0 (t)λ(t) dt < ∞.
6. Differences of positive superharmonic functions
Let D be a Dirichlet regular, Greenian open set, and let u be δ-subharmonic on D.
If µ is the Riesz measure for u, then µ can be written minimally as a difference µ+ −µ−
of two positive measures on D. For all r ∈ ]0, R[ (where R = 1 if n = 2, R = ∞ if
n > 3), we put
+
λ+
D (x, r) = µ (B D (x, r)),
+
ND
(x, r) = pn
Z
r
0
t1−n λ+
D (x, t) dt,
+
−
and similarly for µ− . We say that u(x0 ) is finite if ND
(x0 , ·) and ND
(x0 , ·) are both
finite-valued, in which case it follows from (2) that u is the difference of two superharmonic functions which are finite at x0 . If u(x0 ) is finite, we define the characteristic
TD of u at x0 by
+
TD (u, x0 , r) = LD (u+ , x0 , r) + ND
(x0 , r) − u(x0 )
for each regular value of r.
We use TD to characterize those δ-subharmonic functions on D that can be written as a difference of two positive superharmonic functions, and thus generalize [5]
Theorem 7.42, which deals with the case where D is a ball.
Theorem 7. Let D be a Dirichlet regular Greenian domain, and let u be δsubharmonic on D.
(i) If u = u1 − u2 is the difference of two positive superharmonic functions on
D, and u(x0 ) is finite, then TD (u, x0 , ·) is an increasing function such that 0 6
TD (u, x0 , r) 6 u2 (x0 ) for all regular values of r, and there is a convex function ϕ
such that TD (u, x0 , ·) = ϕ ◦ τ.
(ii) Conversely, if u(x0 ) is finite and TD (u, x0 , ·) is bounded above, then u is the
difference of two positive superharmonic functions on D.
15
.
(i) For i ∈ {1, 2}, let µi be the Riesz measure for ui , and put
λiD (x0 , r) = µi (B D (x0 , r)),
i
ND
(x0 , r) = pn
Z
r
0
t1−n λiD (x0 , t) dt
for all r ∈ ]0, R[. Since u1 > 0, it follows from (2) that
1
0 = LD (u1 , x0 , r) + ND
(x0 , r) − u1 (x0 ).
Since µ1 and µ2 are positive and µ = µ1 − µ2 , we have µ+ 6 µ1 and µ− 6 µ2 , so
that
+
1
(x0 , r) = u1 (x0 ) − LD (u1 , x0 , r).
ND
(x0 , r) 6 ND
Furthermore u1 > u+ , so that LD (u+ , x0 , r) 6 LD (u1 , x0 , r). Hence
TD (u, x0 , r) 6 LD (u1 , x0 , r) + (u1 (x0 ) − LD (u1 , x0 , r)) − u(x0 ) = u2 (x0 ),
which establishes the upper bound for TD (u, x0 , r).
Now put v2 = GD µ− and v1 = u + v2 . Then both v1 and v2 are superharmonic,
so that we can apply (2) to both of them and subtract. Thus we obtain
+
−
u(x0 ) = LD (u, x0 , r) + ND
(x0 , r) − ND
(x0 , r).
It follows that
−
−
TD (u, x0 , r) = LD (u+ , x0 , r) + ND
(x0 , r) − LD (u, x0 , r) = LD (u− , x0 , r) + ND
(x0 , r)
= LD (u− , x0 , r) + v2 (x0 ) − LD (v2 , x0 , r) = v2 (x0 ) − LD (v2 − u− , x0 , r).
Let x ∈ D. If u(x) > 0, then v1 (x) > v2 (x) and v2 (x) − u− (x) = v2 (x) = (v1 ∧ v2 )(x).
On the other hand, if u(x) 6 0 then v1 (x) 6 v2 (x) and v2 (x) − u− (x) = v1 (x) =
(v1 ∧ v2 )(x). Hence
TD (u, x0 , r) = v2 (x0 ) − LD (v1 ∧ v2 , x0 , r).
Since v1 ∧ v2 is superharmonic, the characteristic TD (u, x0 , ·) is increasing on the
set of all regular values (by [11] Theorem 1), there is a convex function ϕ such that
TD (u, x0 , ·) = ϕ◦τ (by [11] Theorem 2), and TD (u, x0 , r) > v2 (x0 )−(v1 ∧v2 )(x0 ) > 0
(by [11] Theorem 1).
(ii) Let w1 , w2 be superharmonic functions such that u = w1 − w2 on D. Applying
(2) to each wj and subtracting, we obtain
(13)
16
−
TD (u, x0 , r) = LD (u− , x0 , r) + ND
(x0 , r)
−
−
for all regular values of r. Therefore ND
(x0 , ·) 6 TD (u, x0 , ·), and so ND
(x0 , ·) is
bounded. Hence
Z R
t1−n λ−
D (x0 , t) dt < ∞,
0
−
so that the function v2 = GD µ is superharmonic on D, by Theorem 6. Furthermore,
+
ND
(x0 , r) = TD (u, x0 , r) − LD (u+ , x0 , r) + u(x0 ) 6 TD (u, x0 , r) + u(x0 )
+
for all regular values of r, so that ND
(x0 , ·) is bounded, and hence the function
+
v1 = GD µ is superharmonic on D. It follows that the function h, defined q.e. on D
by h = u + v2 − v1 , can be extended to a harmonic function h on D. Furthermore,
because v1 and v2 are positive,
−
LD (h− , x0 , ·) 6 LD (u− , x0 , ·) + LD (v1 , x0 , ·) = TD (u, x0 , ·) − ND
(x0 , ·) + LD (v1 , x0 , ·)
by (13), so that
LD (h− , x0 , ·) 6 TD (u, x0 , ·) + v1 (x0 )
by [11] Theorem 1. Therefore LD (h− , x0 , ·) is bounded, so that h− has a harmonic
majorant v on D, by [11] Theorem 1. Hence h = (h + v) − v is a difference of two
positive harmonic functions on D, so that
u = h + v1 − v2 = (h + v + v1 ) − (v + v2 )
is a difference of two positive superharmonic functions on D.
!"
. A representation formula for the difference of two positive superharmonic functions on D, follows from the Riesz decomposition theorem and the Martin
representation theorem for differences of positive harmonic functions on Greenian domains given in [3] p. 204.
References
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applications. J. London Math. Soc. 23 (1981), 137–157.
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[3] J. L. Doob: Classical Potential Theory and its Probabilistic Counterpart. Springer, New
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[7] Ü. Kuran: On measures associated to superharmonic functions. Proc. Amer. Math. Soc.
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[12] N. A. Watson: Generalizations of the spherical mean convexity theorem on subharmonic
functions. Ann. Acad. Sci. Fenn. Ser. A I Math. 17 (1992), 241–255.
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Author’s address: Neil A. Watson, Department of Mathematics and Statistics, University of Canterbury, Christchurch, New Zealand, e-mail: N.Watson@math.canterbury.ac.nz.
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